Method of interpreting impedance distribution of an earth formation using precursor logging data provided by a multi-electrode logging array stationary within a borehole

ABSTRACT

The response characteristics of a combination of hole-centered electric logging tools in a variety of borehole conditions can be obtained by means of impedance values as entries of a matrix resulting from measurements over a series of depth increments, utilizing an array of M electrode assemblies of equal incremental spacing positioned on a rigid mandrel. While the array is substantially stationary within the borehole, currents from ech electrode are emitted independently, one at a time, allowing the impedance matrix to be formed by the principle of linear superposition in terms of voltage and voltage difference measurements between adjacent electrodes; the impedance matrix is inverted and used to generate a set of tool responses, which when properly compared to a set of tool responses previously generated by computer simulation and addressable by sets of borehole parameters and calibration factors, allows the corresponding borehole conditions to be deduced from the measured data.

SCOPE OF THE INVENTION

This invention relates to downhole electrical logging methods andapparatus for carrying out same, and more particularly to a method ofsystematically processing precursor logged voltage and currentinformation so as to provide an aid to improving interpretation ofresistivity of an earth formation about the borehole from whence theprecursor data is collected, as via a stationary multielectrode loggingarray operating in a sequential mode, viz., wherein drive currents aresequentially emitted from each electrode assembly, and the desiredprecursor voltage and current data, are measured at the assemblies ofthe array.

In one aspect of the invention, processing of the field data involvesusing surprisingly accurate reciprocal impedance matrices indexed to aset of predetermined, but overlapping depth scan increments, to generatea series of synthetic responses that can be associated with asurprisingly large number of computer-generated arrays. From thesynthetic responses, a comparison can be easily made to the responses ofsimilar arrays in known borehole and formation conditions and--as aconsequence--the true resistivity (Rt) of the formation as well as theresistivity (Rxo) of the invaded zone, can be determined.

In accordance with another aspect of the invention, such comparisoninvolves constraining the reciprocal impedance matrix so as to simulatelateral and vertical changes in formation resistivity akin to thatprovided by a number of modern focused logging arrays but for which aknown data base for comparison has been previously developed.

RELATED APPLICATIONS

Our applications filed concurrent with and related to the subjectapplication and incorporated herein by reference, include the following:

    __________________________________________________________________________    TITLE                           Ser. No.                                                                              FILING DATE                           __________________________________________________________________________    METHOD OF LOGGING AN EARTH FORMATION                                                                          761,122 July 31, 1985                         PENETRATED BY A BOREHOLE TO PROVIDE AN                                                                        (U.S. Pat. No.                                IMPROVED ESTIMATE OF IMPEDANCE DISTRIBUTION                                                                   4,675,611)                                    OF THE FORMATION;                                                             METHOD OF LOGGING AN EARTH FORMATION                                                                          761,126 July 31, 1985                         PENETRATED BY A BOREHOLE TO PROVIDE AN                                                                        (U.S. Pat. No.                                IMPROVED ESTIMATE OF IMPEDANCE DISTRIBUTION                                                                   4,675,610)                                    WITH DEPTH USING A SINGLE CONTINUOUSLY                                        EMITTING CURRENT ELECTRODE AND A MULTI-                                       PLICITY OF POTENTIAL ELECTRODES OF A MOVING                                   LOGGING ARRAY;                                                                METHOD OF INTERPRETING IMPEDANCE                                                                              761,127 July 31, 1985                         DISTRIBUTION OF AN EARTH FORMATION PENE-                                      TRATED BY A BOREHOLE USING PRECURSOR DATA                                     PROVIDED BY A MOVING LOGGING ARRAY HAVING A                                   SINGLE CONTINUOUSLY EMITTING CURRENT                                          ELECTRODE AND A MULTIPLICITY OF POTENTIAL                                     ELECTRODES;                                                                   METHOD OF LOGGING AN EARTH FORMATION                                                                          761,124 July 31, 1985                         PENETRATED BY A BOREHOLE TO PROVIDE AN                                                                        (U.S. Pat. No.                                IMPROVED ESTIMATE OF IMPEDANCE DISTRIBUTION                                                                   4,677,385)                                    WITH DEPTH USING END EMITTING CURRENT                                         ELECTRODES SEQUENTIALLY ACTIVATED AND A                                       MULTIPLICITY OF POTENTIAL ELECTRODES OF A                                     MOVING LOGGING ARRAY;                                                         METHOD OF INTERPRETING IMPEDANCE                                                                              761,125 July 31, 1985                         DISTRIBUTION OF AN EARTH FORMATION OBTAINED                                                                   (U.S. Pat. No.                                BY A MOVING ARRAY USING END EMITTING                                                                          4,677,386)                                    CURRENT ELECTRODES SEQUENTIALLY ACTIVATED                                     AND A SERIES OF POTENTIAL ELECTRODES                                          __________________________________________________________________________

BACKGROUND OF THE INVENTION

From a knowledge of the voltage distribution in earth formationpenetrated by a borehole resulting from imposed current flow in theformation, hydrocarbon saturation of the formation can be determined.Rock matrices are generally nonconductors of electricity. But if theformation is porous and contains fluids, current can be driven throughthe formation, and the voltage distribution along the borehole measured.The impedance of the formation relates to its ability to impede the flowof current through the formation, and is measured in ohms.

The resistivity of the formation also relates to the ability of theformer to impede current flow but is measured not in ohms but in termsof ohm meter² per meter or ohm-meter. That is to say, the resistivity ofa formation is the impedance (in ohms) of a one meter by one meter byone meter cube of the formation when the current flows between oppositefaces of the cube. Resisitivities fall in the range from 0.2 to 1000ohm-meter in most permeable earth formations we are familiar with.

Since the formation to be logged is penetrated by a borehole containinga fluid having a resistivity other than that of the adjacent formation,the obtained apparent resistivity (Ra) can differ from the trueresistivity (Rt) of the formation. That is to say, the presence of theborehole filled with a fluid having a resistivity Rm different from thatof the formation, the fact that the drilling fluid filtrate invades theformation to a limited degree and flushes away formation water and somehydrocarbons to establish a resistivity Rxo again different from that ofthe formation; and the fact that the measuring electrodes may cross intoadjacent formations, all perturb the final results.

Certain electrical logging methods overcome such perturbations becauseof novel borehole conditions. For example, conventional resistivity logs(non-focused logs), provided by conventional electrical survey (ES)tools, provide good true resistivity estimates only in thick homogenousbeds having porosities greater than 15 per cent. For thinner bedconditions, such tools can provide reliable results if filtrate invasionis shallow, if the true resistivity is low to moderate and if theresistivity of flushed zone is equal to or less than the trueresistivity to be measured.

Additional more advanced logs have been developed to concentrate onenhancing the focusing properties of the electrical tools to overcomethe abovementioned perturbations. For example, families of resistivitytools have been developed in the last quarter century which use focusingcurrents to control the paths taken by the measuring current: among suchtools, are the focusing logging tools including the spherically focusedtool. Such tools use currents supplied to special electrodes on the tooland their readings are less affected by borehole conditions and thepresence of adjacent beds.

But to an essential degree both types of logs have not been flexibleenough under the varying borehole conditions encountered in today'sproduction fields, on land or at sea. For example, conventional ES logsare too broadly structured to provide a way for a user to determinefocusing response of electrical tools independent of electrodearrangement. Conversely, focused electrical logs are too strictlyformulated to provide such independent results. That is, insufficientmeasurements are provided to yield results of focusing characteristicsbeyond that of the orginal configuration. In addition, calibrationfactors for deep and shallow focused tools appear to be chosen so thattheir responses are equal to the true formation resistivity in uninvadedformations having formation/mud resistivity contrasts in a range of 10/1to 100/1 normalized to an eight-inch borehole. Hence in order for theuser to have the option to test different focusing responses independentof electrode arrangement, he had to develop an entirely differentlogging method.

One such proposal is set forth in U.S. Pat. No. 3,076,138 for"Electrical Logging", R. B. Stelzer, in which a multiple electrode arraytool is used to provide voltage and current measurements that can bearrranged in matrix format within a digitial computer, as a function ofdepth along the borehole.

In the patent, the genesis of the matrix format is described in terms ofa 2×3 array divided into six submatrices, one of which is a square arraywhose entries are surprisingly found to be independent of any laterelectrode arrangement to be synthesized. The above-mentioned squaresubmatrix has rows which can be filled with raw field data values e.g.,to identify the voltage at a common depth position and columns of valuesthat can identify voltage response at a series of voltage electrodes(including the current emitting electrode), as a function of commoncurrent electrode location.

It is believed that this proposal is the first to recognize thatelectrode logging data (viz., current, resistance and voltage) could becombined in such a matrix format.

In field operations, a bottom mounted current electrode is continuouslyenergized as the sonde is moved through the borehole. Absolute voltagemeasurements at each of the series of uphole pickup electrodes(including the current emitting electrode) are sensed and recorded withrespect to a remote uphole voltage reference electrode. A return currentelectrode is also mounted on the bridle of the tool, suitably locatedfrom the other electrodes, and the current from the emitting electrodeis also measured and recorded. By dividing the measured absolutevoltages by the corresponding measured current in accordance with Ohm'sLaw in matrix format, a resistance matrix R between arbitrary syntheticvoltage and current values can be established. (Henceforth, matrixquantities will be underscored.) In principle, such a resistance matrixis suitable for synthesizing substantially the responses of conventionalelectric logging tools by manipulation of the matrix elements. Suchoperations as explained in more detail below, are most important in theeffectuation of the scheme in accordance with the proposal because ofits basic property of allowing the synthetic currents to be uniquelydetermined from the corresponding voltages, or vice versa.

Also of importance in the practical implementation of the proposal isthe recognition that it will generally be necessary to solve systems ofequations involving the aforementioned submatrix, or what is equivalentto accurately computing the mathematical inverse of the submatrix tosimulate responses of modern focused tools. That is to say, the solutionof the reciprocal of such submatrix will be generally necessary for thesynthesis of modern focused tools and especially for the synthesis ofnew and heretofore unknown electrode combinations requiring arbitraryvoltage-current relationships. Thus, the above proposal is applicableonly to those situations for which it is possible to produce the inverseof the submatrix with sufficient accuracy. But experience has indicatedthat in many field applications such results are not possible. Theproblem has to do with the numerical constraints imposed by themeasurement process which ultimately result in finite limited precisionof the voltage measurements, and has appeared with regularity in thosefield situations for which the formation to mud resistivity contrast isgreater than 100 to 1 (viz., in situations where salty drilling fluidsare used; where the uninvaded formation is of low porosity; and wherethere is moderate to high hydrocarbon saturation). It is believed thefailure of the proposal to provided accurate results, has to do with thefact that in such high contrasts, the potential tends to change veryslowly from electrode to electrode. Thus it has been impossible topreserve the required precision to accurately resolve the gradualvariation involved. As a consequence, in subsequently manipulatingmatrix potential values, such as where floating point calculationsspecify differences in the potential between adjacent electrodes, themethod of the proposal breaks down.

More recently, a second proposal has been put forth in U.S. Pat. No.4,087,741 for "Downhole Geoelectric Remote Sensing Method", I. R. Mufti,in which a multiple electrode array tool is described for the detectionof lateral resistive anomalies remote from the borehole. Typically, suchanomalies are salt domes. This system uses the superposition principleto achieve synthesis of various four (4) electrode tools in the mannerof extremely ultra long spaced electric logging tools (ULSEL)--see R. J.RUNGE ET AL., "Ultra-long Spaced Electric Log (ULSEL)", THE LOG ANALYST,Vol. 10, No. 5, September-October, 1969.

More specifically in this proposal, a center mounted current electrodearray (viz., a current electrode with voltage sensing electrodesdisposed sysmmetrically above and below the current electrode) isdisposed on a bridle of ultra-long length. The current electrode iscontinuously energized at a low frequency as the bridle is moved throughthe borehole. Voltage differences between adjacent sensing electrodesabove and below the current electrode are measured and recorded. Theexclusive purpose of the tool: to synthesize various long-range,four-electrode tools for the detection of lateral anomalies. Since thevoltage sensing electrodes are non-uniformly spaced, and sincequantities related to the driving point resistance (i.e., the drivingpoint impedance at the current emitting electrodes) are not measured,the proposal does not result in the type of matrix formulation providedby either the first-mentioned proposal or that provided by the presentinvention. That is to say, while the second proposal will allowcalculations of potentials at given electrodes in presence of certainarbitrary currents at other electrodes, it will not allow the inversecalculations, i.e., the calculation of current at a given electrodeposition for given potentials at other electrode positions via ameasured impedance matrix. It therefore cannot be used either inprinciple or in practice to synthesize other types of logging tools ofinterest in general.

RELATED APPLICATION

In our co-pending application for "METHOD OF LOGGING AN EARTH FORMATIONPENETRATED BY A BOREHOLE TO PROVIDE AN IMPROVED ESTIMATE OF IMPEDANCEDISTRIBUTION OF THE FORMATION", filed concurrently herewith, there isdescribed in detail how to use the improved impedance matrix as providedby a stationary electrode array using in sequence current electrodes ofthe array in conjunction with a mutiplicity of potential electrodes.These steps included a substep of determining resistivity values thathave been formulated using synthetic voltages generated from theprecursor measured voltage and current values. Since the syntheticpotential values are based in part on potential differences betweenadjacent pairs of electrode assemblies, the former are surprisinglyaccurate under all types of borehole conditions including those whereresistivity contrasts of the formation and drilling mud are over 100:1.

However, there is a further need to accurately interrelate thedifferences in array responses other than by generation of a series ofresistivity values to take into account--in a systematic manner--thevariation in responses of such arrays as a function, for example, ofdrilling mud filtrate invasion and differing resistivity contrasts inand around the borehole penetrating the formation under survey.

SUMMARY OF THE INVENTION

In accordance with the present invention, a method is disclosed forsynthesizing the true response characteristics of a combination ofdifferent holecentered electric logging tools in a variety of difficultborehole conditions as provided by (1) determining impedance values ofan earth formation penetrated by a borehole filled with a drilling mudof resistivity (Rm), and (2) selectively manipulating the impedancevalues as impedance entries of a series of reciprocal matrices so as tosynthesize operations of different hole-centered tools over anassociated depth increment with surprising accuracy. Each of thereciprocal impedance matrices is associated with a reciprocal matrixgather indexed to one of a series of finite, overlapping depth scanincrements of the formation measured along the borehole. Each scanincrement is dependent on the array length L of the electrode array todefine shallow and deep depth markers as well as being centrally indexedto the depth in the borehole of a mid-central electrode assembly of thearray at the time of data collection.

In more detail, in order to provide a true indication of the formationresistiveity (Rt) even though the formation is interspaced from theborehole by an invaded zone of resistivity (Rxo) of unknown lateralextent due to drilling mud filtrate invasion, the following steps arecarried out in sequence:

(i) an array of hole centered M electrode assemblies of equalincremental electode spacing "a" is first calibrated to obtain sets ofcalibration factors normalized to known current response and voltageinitiation patterns in a known resistivity zone of response, saidelectrode assemblies having a known internally ordered numbering index,and each comprising a current and potential electrode, said sets ofcalibration factors each being addressable as a function of boreholeconditions including said difficult borehole conditions as well as bysynthetic computer focused array type;

(ii) next the array is positioned in the borehole, wherein the absolutedepth of at least one electrode assembly is continuously known withrespect to a predetermined depth datum level measured from the earth'ssurface;

(iii) while the array is substantially stationary within the borehole,current is injected from each of said electrode assemblies--one at atime--into the adjacent formation;

(iv) at each occurrence of current injection, the absolute potential ata common electrode assembly as well as the potential differences betweenadjacent pairs of assemblies, are measured as a function of internalindexing numbers of active current and potential sensing electrodes ofthe assemblies used in the measurements;

(v) after the array is advanced along the borehole an increment distanceequal to the electrode assembly spacing "a" between adjacent assemblies,steps (iii) and (iv), are repeated while, again the array issubstantially stationary within the borehole:

(vi) then steps (iii)-(v) are re-repeated until the collection of allcurrent and potential data has been completed;

(vii) then impedance values are calculated from the measured absoluteand difference potential values and their associated injection currents,each value being indexed to said known internal indexing numbers ofactive current and potential electrodes used in the measurements;

(viii) next the impedance values are reindexed to form impedance entriesof a series of independent, but overlapping matrix gathers ΔZ, eachgather being associated with a predetermined segment of said formationequal in vertical extent to M logging stations, and comprising M×Mimpedance entries where M is the largest number of the numbering indexof the electrode assemblies comprising said array and in which the ratioof the number of difference impedance entries to absolute entries isabout M-1:1,

(ix) then the entries of each matrix gather ΔZ are inverted to form thereciprocal matrix gather thereof Δz⁻¹ in accordance with conventionalmatrix inversion techniques;

(x) computer focused response parameters are next generated using thereciprocal matrix gather Δz⁻¹ of step (ix) in conjunction with the samecurrent response and voltage initiation patterns of step (i);

(xi) finally sets of calibration factors of step (i) are searched untilthe product of a particular set of calibration factors and the responseparameters of step (x) for all synthetic tool array types is essentiallya constant whereby the difficult borehole condition is deduced even inthe presence of high true resistivity to mud resistivity contrasts andirrespective of the fact that synthetic sets of potential patterns havebeen used as initiators of the subsequently generated computer focusedresponse parameters.

DESCRIPTION OF DRAWINGS

FIG. 1 is a partial side elevation of an earth formation penetrated by aborehole containing an electrical logging array and illustrating amethod for processing estimates of the impedance distribution of theearth formation about the borehole including a form of the invention inwhich the precursor current and voltage values for the estimates aregained via a logging array that is stationary in the borehole as theprecursor values are generated at a series of logging stations indexedto depth positions denoting fixed depths in units of electrode spacing;

FIG. 2 is a detail of a typical electrode assembly of FIG. 1illustrating constructional details of the current and potentialelectrodes comprising that assembly;

FIG. 3 is a section taken along the line 3--3 of FIG. 1 illustrating howdetected measurements provided by the array can be used to deduce theresistivity characteristics of the formation as a function of offsetdistance even though the formation has undergone invasion by thedrilling mud;

FIG. 4 is a schematic representation of the method of the presentinvention for systematically indicating the impedance distribution ofthe adjacent earth formation using a modified 5-electrode assembly arrayin which the indexing features of the displays are detailed;

FIG. 5 is an enlarged detail of the final displays of FIG. 4;

FIGS. 6 and 7 are schematic representations of uphole and downholecircuit elements for carrying out the method of the present invention;

FIG. 8 is an equivalent artificial network simulating an earth formationto illustrate the unique correspondence of the admittances of thatnetwork and the reciprocal impedance matrix ΔZ⁻¹ of the presentinvention;

FIG. 9 is a plot of pseudo-geometric factor G for different computerfocused arrays as a function of diameter of invasion (Di) using anormalized current response and voltage initiation pattern and a seriesof known borehole conditions:

FIGS. 10-28 are plots of calibration factors versus resistivitycontrasts illustrating how sets of such factors are unique to a selectedborehole condition so as to allow detection thereof in accordance withthe method of the present invention.

DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

In the description that follows like elements are marked throughout theSpecification and Drawings with the same reference numerals. Thedrawings are not to scale and certain features of the invention may beexaggerated in scale or operation in the interest of clarity orconciseness.

Referring to FIG. 1, a borehole 8 is shown penetrating a subsurfaceformation 9. Assume the borehole 8 contains a drilling mud 10. Purposeof the drilling mud 10: to carry chips away from the drill bit duringdrilling of the borehole 8 as well as to prevent collapse of theborehole 8 as during drilling and production operations. Also suspendedwithin the borehole 8 is a conductor cable 12 of conventionalconstruction, i.e., multiple strands of flexible steel interlaced with aplurality of electrical conductors. The cable 12 is suspended withinborehole 8 via pulley 13 attached to derrick 14 at the earth's surface15 and thence to hoisting unit 16. Purpose of hoisting unit 16: to reelin or play out cable 12 within the borehole 8.

At the earth's surface 15, signals on the electrical conductors of thecable 12 pass through a slip ring assembly (not shown) on the hoistingunit 16 and thence to a controller-processor circuit 17 within van 18,as via conductor 19. Downhole, such signals originate at and are afunction of the operational characteristics of electric logging array 21of the present invention.

ELECTRIC LOGGING ARRAY 21

Downhole, the cable 12 is attached to logging array 21 via a threadedplug 22. Above the plug 22 is a uphole centralizer 23. Below the array21 is downhole centralizer 24. The centralizers 23 and 24 are similar inconstruction and each includes a series of spring loaded arms 25 whichare biased radially outward into contact with sidewall 5 of the borehole8. The arrangement of the centralizers 23 and 24 is similar tocentralizers used in various types of logging and inspection tools andis for the purpose of locating the logging array 21 coincident with theaxis of symmetry A--A of the borehole 8.

Logging array 21 is cylindrical in construction having a supportstructure defined by a mandrel 26. The mandrel 26 has an outer surface26a. At the surface 26a are a series of electrode assemblies having aninternal numbering index E₁, E₂ . . . E_(N+1) . . . E_(M). In order thatthe adjacent assemblies be electrically isolated one from the other, aninsulating material is fitted between each electrical assembly E₁, E₂ .. . E_(M) and the outer surface 26a of mandrel 26. The array 21 isdivided into three sections: (i) Uphole section 27 within which residereturn current electrode 28 and reference potential electrode 29, thepurpose of such electrodes 28 and 29 being to complete the current loopand to normalize potential measurements, respectively, as conventionalin the electric logging art. If desired, the metallic strands of thecable 12 may also be used as the reference electrode as well as thereturn current electrode, as is also conventional in that art: (ii) Midsection 30 that contains threads 31 at its upper end for engagement withuphole section 27. Interior of midsection 30 are a series of slavecontrol and measuring elements to be described in detail hereinafter,under control of master circuitry of controller-processor unit 17 at theearth's surface 15; and (iii) Downhole section 33 on which reside theseries of electrode assemblies E₁, E₂ . . . E_(M), previously mentioned.

FIG. 2 shows the construction of a typical electrode assembly E₁, E₂ . .. or E_(M) in more detail.

As shown, each such assembly includes a current electrode member 34 anda potential sensing electrode 35, both of annular construction that fitthe two ends of insulating member 42. Between the members 34 and 35 isan insulating lip 41 which is part of member 42, and serves to keepelectrodes 34 and 35 from being in physical contact with one another.Members 34, 35 and 42 are located about outer surface 26a of mandrel 26concentric of the axis of symmetry B--B.

Note when members 34 and 35 are attached in the manner depicted in FIG.1 so as to function as current and potential electrodes, respectively,that they have widths in the vertical direction that are small comparedto the axial spacing "a" between the electrode assemblies, and, inaddition, they are closely nested relative to each other (that is, thethickness of the lip 41 of FIG. 2 is small compared to the axial spacing"a") that for purposes of axial resolution, their axial position isequivalent to a single fixed depth in FIG. 1.

With regard to the internal numbering order of the electrode assemblies,note that the shallowest assembly is assembly E₁ and the deepestassembly is E_(M) and that the depth of any electrode assembly E₁, E₂ .. . E_(M) can be determined based on

    Depth=d.sub.k +(p-1)a,

where p is 1, 2 . . . M; d_(k) is the absolute depth of the shallowestelectrode assembly of the array; and M is the deepest electrode assemblyof the array.

OPERATIONS

Briefly, in accordance with the present invention, the purpose oflogging array 21 is to provide an impedance distribution of theformation 9 such that such values preserve a one-to-one relationshipbetween any subsequent combination of voltages and currents that couldpossibly be used to synthesize the response characteristics from anynumber of different electrical tools under a variety of difficult anddifferent borehole and formation conditions. To achieve such a goal, thelogging array 21 is operated in the manner depcited in FIG. 1 so thatcurrent and potential values are systematically collected as function ofdepth. Next, controller-processor circuit 17 at the earth's surface 15,selectively manipulates such values to form entries of a series ofmatrix gathers, each gather being exceedingly useful in synthesizingoperations of different logging tools within the borehole 8 penetratingthe formation 9. Ultimate purpose: to provide a true indication ofcharacteristics of formation 9 even though that formation may have beeninvaded by drilling mud filtrate in and around the borehole to anunknown extent as in the manner of FIG. 3, whether or not synthetic setsof potential or current vectors are later used as initiators of thesimulated computer focused tool arrays.

That is to say, as shown in FIG. 3 note that the borehole 8 to be loggedvia logging array 21 can contain drilling mud 10 of resistivity Rm; thatthe drilling mud 10 can build up a mud cake 6 at the sidewall 5 of theborehole 8; and that the mud filtrate can invade the formation 9 adistance D from the center line of the borehole 8 thereby creatingdifferent resistivity levels in and around the borehole 8, viz., a mudcake zone of resistivity Rmc; a flushed zone 7 of resistivity Rxo, and atransition zone 11 of resistivity that may vary between the resistivityof the flushed zone and of the formation 9, viz., between Rxo and Rt andfor purposes of discussion is called Rapp. In order to indicate thelateral resistivity changes vis-a-vis the drilling mud 10, the flushedzone 7, and the formation 9, not only must the logged current andpotential values as provided by logging array 21 (as a function ofdepth) be systematically collected and indexed, but the impedance valuescalculated from these values must also be accurately indexed so thatafterward, viz., say after logging has been completed, they can beselectively manipulated to provide different degrees of lateralresolution irrespective of whether or not the response initiators forsuch manipulations, are artificial sets of current or potential values.In that way, the lateral resistivity changes mentioned above can beeasily determined.

During collection of data in accordance with FIG. 1, the logging array21 remains stationary as the current electrode of each assembly E₁, E₂ .. . E_(M) is sequentially activated. Current injection typically beginswith assembly E₁ and proceeds in ordered sequences through the remainingassemblies E2, E3, etc., and ends with assembly E_(M). Potentialmeasurements occur at all the electrode assemblies E₁, E₂ . . . E_(M) ina similar sequential pattern beginning initially at assembly E₁ andproceeding downward to assembly E_(M). Next, the logging array 21 isrolled up or down the borehole one logging station, say where assemblyE₁ is rolled down from station d₁ to station d₂, and the collectionprocess is repeated. Result: a series of current and potential valuesare systematically collected as a function of depth for latermanipulation as set forth below. But note that current and potentialvalues to be manipulated only have formation integrity if they allrelate to the same stationary collection local. That is to say, valuesmust be indexed to a particular stationary depth scan interval (equal toM logging stations) that prevents intertwining of like values ofdifferent depth scan intervals, in a manner explained in more detailbelow.

FIG. 4 illustrates how systematic collection and indexing occurs duringoperations. for description purposes, it is assumed that the number ofelectrode assemblies comprising the logging array 21 has been greatlycurtailed, say scaled down from the large array of FIG. 1 to a5-electrode array comprising electrode assemblies E₁, E₂ . . . E₅, andthe 5-electrode array 21 is sequentially rolled down only a selectednumber of logging stations. That is, only four separate collectioncycles, viz., cycles 1, 2 . . . 4 for logging positions A, B, C, and Dwill be described in detail. But at each logging position there will begenerated sufficient precursor current and potential values to form a5×5 matrix gather and it is from that standpoint that the 5×5 gathermatrices of the present invention are presented. In associating themeasured potential and current values into 5×5 gather matrices, it willbecome evident that potential values collected above the currentelectrode will provide matrix entries above its diagonal, while thosecollected below will give entries below its diagonal. In this regard,the ordinate of the plot in FIG. 4 is in units of depth and the abscissais in units of incremental time or cycles 1, 2 . . . 4. Spacing betweenthe assemblies E₁ . . . E₅ is equal to spacing factor "a", as is thedistance between logging station d₁, d₂ . . . d₈. At each loggingposition A, B, C, and D, note that the array 21 is stationary duringcollection of the potential, phase and current values. Thereafter, thearray is rolled downward to the next logging position and the collectionprocess is repeated. Movement of the array 21 occurs because of reelingout of cable 12 via hoisting unit 16. The collected values aretransmitted uphole via the cable 12 and at the earth's surface 15 thencefrom the hoisting unit 16 to the controller-processor circuit 17.Because of the large mass of data indexing of the logged values israther important and dependent upon the absolute as well as relativedepth positions of the emitting current electrode as well that of thepotential measuring electrodes comprising the electrode assemblies E₁,E₂ . . . E₅.

For example, for measurements taken when array 21 is at position A inFIG. 4, the current electrode of electrode assembly E₁ at depth markerd_(k) coincident with logging station (d₁), is energized. For the array21 each measuring cycle 1, 2 . . . 4 requires the collection of thefollowing analog values: (1) 4-potential difference values, (2) a singleabsolute potential values associated with a common electrode assembly,(3) a single current intensity value and (4) a pair of control valuesrelated to indicating phase distortion, i.e., indicating distortion viacalculated time difference values between the current at energizingelectrode and the absolute potentials at the corresponding potentialsensing electrode and the most remote potential sensing electrode of theassemblies vis-a-vis the position of the emitting current electrode.These values are transmitted uphole via cable 12 and thence fromhoisting unit 16 to controller-processor circuit 17 for storage; andmanipulation in accordance with the method of the present invention.

In order to assure that addresses of the collected current and potentialvalues are complete, the following indices are made of record, vis-a-visthe collected current and potential values, viz.: (i) by depth markersd_(k), d_(k) +a . . . d_(k) +8a where the factor "a" is the incrementalspacing between electrode assemblies E₁, E₂ . . . E₅ and d_(k) is theabsolute depth of the electrode E₁ at the start of data collection,viz., when the array is positioned at location A associated withincremental logging time unit 1; (ii) by consecutive numbered electrodelogging stations (d₁, d₂, d₃ . . . d₈) associated with the entirelogging operation as where the relative position of each station is ofinterest; (iii) by scan depth station number (Sd₁, Sd₂, etc.) associatedwith the depth of the mid-central electrode assembly, viz., d_(k) +2a,d_(k) +3a, d_(k) +4a, and d_(k) +5a, respectively, at the occurrence ofthe four depicted collection cycles. These values can be indexed in anumber of different formats as the data is collected, typical of whichbeing displays 46, 47, 48, 49, and 50, and then being re-indexed inmatrix gather format as set forth in display 51. It should be furthernoted that the displays 46,47 . . . 50 have a further annotation tag:viz., that the depicted values forming each such display must be furtherindexed to indicated the depth of the current electrode assemblies usedas the current emitter during each of the collection cycles 1, 2, 3 . .. which give rise to display 46, 47 . . . 50, respectively. Suchannotation system can also be carried over into the re-indexed matrixgather display 51 of the impedance values associated with thesemeasurments, as explained below.

That is to say, assume that absolute depths of the numbered loggingstations are known; so that when the array 21 is located at position Athen the electrode assemblies E₁, E₂, E₃ . . . will be associated withthe internal numbering index 1 . . . 5 of consecutive order; hence, whenthe current electrode of electrode assembly E₁ is at depth d_(k)coincident with logging station d₁ and measurement at the associatedelectrode assemblies taken, then the absolute and differential potentialvalues and current intensity would be indicated by the followingquantities: ##EQU1##

Note with respect to the indices for the absolute potential that thefirst subscript relates to the internal index number of the 5-electrodearray at which the potential measurement occurs and the second subscriptidentifies the internal index number of the current electrode undergoingenergization while the address in parenthesis relates absolute depthfrom say the earth's surface 15 to the position of the current electrodeundergoing activation. In regard to the last-mentioned address tag, thelogging station of the current electrode, viz., logging station (d₁),could be used also as substitute since absolute depth can be latercalculated.

Note that the potential differences are measured between the pairs ofelectrode assemblies, i.e., between electrode assemblies 1 and 2; 2 and3; 3 and 4; etc., these values are also indexed in a similar manner asabove. That is, in accordance with following:

    ΔV.sub.i,1 (d.sub.k) where i=2 . . . 5.

Note that the above, the first subscript relates the position of thedeeper of each pair of electrode assemblies and assumes that thenormalizing value for forming the difference potential value relates tothe descending ordered electrode assembly. That is, the value

    Δ.sub.2,1 (d.sub.k)

indicates that the potential difference is measured between thepotential electrodes of assemblies E₁ and ₂ internally numbered as 1 and2, respectively, and that the current electrode energized has aninternal index number of 1, at a depth (d_(k)), while the value

    ΔV.sub.4,1 (d.sub.k)

indicates that the potential difference is measured between assembliesE₃ and E₄ having internal index numbers 3 and 4 in FIG. 4, with thecurrent emitter being associated with the internal numbered assembly 1,at the marker depth d_(k). Note that depiction of the aforementionedvalues as set forth above, comprises entries of columns 46a, 46b, and46c of display 46, the current intensity being the sole entry of column46c while the time measurements [T₁ (d_(k)) and T₅ (d_(k))] associatedwith indicating phase distortion, if any, are set forth as the entriesof column 46d.

With the array stationary, the next step of the method in accordancewith the invention is to activate the current electrode of electrodeassembly E₂ identified as internal index number 2, and measure theabsolute voltage as assembly E₁ and the potential difference valuesbetween adjacent pairs of electrode assemblies E₁ . . . E₅, as well asdthe current intensity. That is provide the values

    V.sub.1,2 (d.sub.k +a)

    ΔV.sub.i,2 (d.sub.k +a)

where i=2, 3 . . . 5; and

    J.sub.2 (d.sub.k +a).

In addition, a pair of control values ]T₂ (d_(k) +a) and T₅ (d_(k) +a)]that indicated the degree of phase distortion are also measured. Thesevalues occupy entries of columns 47a, 47b, 47c, and 47d of display 47.

Thereafter, the current electrodes having internal index numbers 3, 4,and 5 are sequentially activated and at each time of occurence, theabsolute voltage at assembly E₁, the potential difference betweenadjacent pairs of electrode assemblies, the current intensity and thepair of phase control values are measured, viz., ##EQU2## where i=2, 2,4, and 5.

The above described values occupy entries of columns 48a, 48b . . . 48dof display 48; columns 49a, 49b . . . 49d of display 49; and columns50a, 50b . . . 50d of display 50.

Table I, below, sets forth the measurements with logging operations A,B, C, and D associated with displays 46 . . . 50 in tabular form forgreater clarity.

                  TABLE I                                                         ______________________________________                                        C46a    C46b      C46c     C46d                                               ______________________________________                                        V.sub.1,1 (d.sub.1)                                                                   ΔV.sub.2,1 (d.sub.1)                                                              J.sub.1 (d.sub.1)                                                                      T.sub.1 (d.sub.1)                                          ΔV.sub.3,1 (d.sub.1)                                                                       T.sub.5 (d.sub.1)                                          ΔV.sub.4,1 (d.sub.1)                                                    ΔV.sub.5,1 (d.sub.1)                                                                              = DISPLAY 46                                ______________________________________                                        C47a    C47b      C47c     C47d                                               ______________________________________                                        V.sub.1,2 (d.sub.2)                                                                   ΔV.sub.2,2 (d.sub.2)                                                              J.sub.2 (d.sub.2)                                                                      T.sub.2 (d.sub.2)                                          ΔV.sub.3,2 (d.sub.2)                                                                       T.sub.5 (d.sub.2)                                          ΔV.sub.4,2 (d.sub.2)                                                    ΔV.sub.5,2 (d.sub.2)                                                                              = DISPLAY 47                                ______________________________________                                        C48a    C48b      C48c     C48d                                               ______________________________________                                        V.sub.1,3 (d.sub.3)                                                                   ΔV.sub.2,3 (d.sub.3)                                                              J.sub.3 (d.sub.3)                                                                      T.sub.3 (d.sub.3)                                          ΔV.sub.3,3 (d.sub.3)                                                                       T.sub.5 (d.sub.3)                                          ΔV.sub.4,3 (d.sub.3)                                                    ΔV.sub.5,3 (d.sub.3)                                                                              = DISPLAY 48                                ______________________________________                                        C49a    C49b      C49c     C49d                                               ______________________________________                                        V.sub.1,4 (d.sub.4)                                                                   ΔV.sub.2,4 (d.sub.4)                                                              J.sub.4 (d.sub.4)                                                                      T.sub.4 (d.sub.4)                                          ΔV.sub.3,4 (d.sub.4)                                                                       T.sub.1 (d.sub.4)                                          ΔV.sub.4,4 (d.sub.4)                                                    ΔV.sub.5,4 (d.sub.4)                                                                              = DISPLAY 49                                ______________________________________                                        C50a    C50b      C50c     C50d                                               ______________________________________                                        V.sub.1,5 (d.sub.5)                                                                   ΔV.sub.2,5 (d.sub.5)                                                              J.sub.5 (d.sub.5)                                                                      T.sub.5 (d.sub.5)                                          ΔV.sub.3,5 (d.sub.5)                                                                       T.sub.1 (d.sub.5)                                          ΔV.sub.4,5 (d.sub.5)                                                    ΔV.sub.5,5 (d.sub.5)                                                                              = DISPLAY 50                                ______________________________________                                         LEGEND:                                                                       (d.sub.1) = (d.sub.k); (d.sub.2) = (d.sub.k + a); (d.sub.3) = (d.sub.k +      2a); (d.sub.4) = (d.sub.k + 3a); and (d.sub.5) = (d.sub.k + 4a).         

From the above-denoted measured values of potential and currentintensity, their corresponding ratio, i.e., measured values associatedwith the same set of electrical variables depicted within each display46, 47, 48 . . . 50 can be determined using the following indices andequations, viz., ##EQU3## where i=2, 3 . . . 5.

Thereafter, the above results associated with the above equations can bere-indexed in matrix gather format to generated the display 51 aspreviously mentioned. Note in this regard that the matrix entries setforth in the display 51 preserve the one-to-one relationship of thecurrent and potential values collected with the logging array 21 at thedifferent logging positions in FIG. 4. These entries are set forth intabular form in Table II and are easily annotated from the precursorpotential and current values found in Table I. In comparing the entriesof Tables I and II, note that the scan depth (sd₁) of the depictedmatrix gather is coincident with depth marker (d_(k) +2a).

    ______________________________________                                        C1          C2       C3      C4       C5                                      ______________________________________                                         ##STR1##                                                                             .sup. Z.sub.1,1                                                                       .sup. Z.sub.1,2 (d.sub.2)                                                              .sup. Z.sub.1,3                                                                     .sup. Z.sub.1,4 (d.sub.4)                                                            .sup. Z.sub.1,5 (d.sub.5) R1                    (d.sub.1)        (d.sub.3)                                                    ΔZ.sub.2,1                                                                      ΔZ.sub.2,2 (d.sub.2)                                                             ΔZ.sub.2,3                                                                    ΔZ.sub.2,4 (d.sub.4)                                                           ΔZ.sub.2,5 (d.sub.5) R2                   (d.sub.1)        (d.sub.3)                                                    ΔZ.sub.3,1                                                                      ΔZ.sub.3,2 (d.sub.2)                                                             ΔZ.sub.3,3                                                                    ΔZ.sub.3,4 (d.sub.4)                                                           ΔZ.sub.3,5 (d.sub.5) R3                   (d.sub.1)        (d.sub.3)                                                    ΔZ.sub.4,1                                                                      ΔZ.sub.4,2 (d.sub.2)                                                             ΔZ.sub.4,3                                                                    ΔZ.sub.4,4 (d.sub.4)                                                           ΔZ.sub.4,5 (d.sub.5) R4                   (d.sub.1)        (d.sub.3)                                                    ΔZ.sub.5,1                                                                      ΔZ.sub.5,2 (d.sub.2)                                                             ΔZ.sub.5,3                                                                    ΔZ.sub.5,4 (d.sub.                                                             ΔZ.sub.5,5 (d.sub.5) R5                   (d.sub.1)        (d.sub.3)                                            ______________________________________                                         LEGEND:                                                                       (Sd.sub.1 = d.sub.3 = d.sub.k + 2a) Logging Stations d.sub.1, d.sub.2,        d.sub.3, d.sub.4 and d.sub.5 are equal to d.sub.k ; d.sub.k + a; d.sub.k      2a; d.sub.k + 3a; and d.sub.k + 4a, respectiv ely.                            Z = Absolute impedance;                                                       ΔZ = differential impedance.                                            ##STR2##                                                                 

In Table II note that the entries addressed by row, viz., row R1, R2,R3, R4, or R5, and by column, viz., column C1, C2, C3, C4, or C5, areobained by direct measurement and not by means of reciprocity, and thatannotation as to depth of survey scan is in accordance with

    Depth scan depth=d.sub.k +Na, M=2N+1.

That is to say, for the instance where d_(k) +Na is equal to d_(k) +2afor the matrix of Table II, the scan gather depth is the midpoint of theextreme depths d_(k) and d_(k) +4a and is thus is defined by the aboveequation.

As to a general relationship of the rows and columns of the matrix ofTable II, it is noted that the diagonal and supradiagonal entries (i.e.,entries above the diagonal) require entries from Tables I for thosesituations for which the active current electrode for the array neverlies above the electrode assemblies of the array never which potentialsare measured. It follows that the subdiagonal elements of the matrix(those entries below the diagonal) require entries from Table II forthose situations for which the active current electrode for the arraynever lies below the electrode assemblies at which potentials aremeasured when the array 21 is in position A.

Thereafter, the logging array 21 of FIG. 4 is lowered in sequence tologging positions B, C, and D and the above-described collection processis repeated. Result: each location B, C, or D can be associated withsets of precursor current, potential and phase displays, generallyindicated at 52, 53, 54, which in turn result in the generation ofassociated matrix gathers 55, 56, and 57 in a mannner similar to thatpreviously described wherein each determined matrix gather 55 . . . 57preserves the one-to-one relationship for the current and potentialvalues collected during positioning of the logging array at positions B,C, or D.

FIG. 5 illustrates gather 51 associated with logging positin A, as wellas gathers 55, 56, and 57 of positions, B, C, and D, respectively, inmore detail.

As shown, the entries comprising each display 51, 55, 56, or 57 are seento have the following common characteristics: (i) they are related toprecursor measured values within the formation adjacent to the borehole;(ii) they are indexed to a particular zone within the earth by amid-point depth marker coincident with the mid-central electrodeassembly of the array, and by two depth values that define the bounds ofthe zone of interest at the time of collections, and (iii) they are alsoindexed to a particular scanning depth station number associated withthe depth of the mid-central electrode assembly. That is to say, each ofthe entries forming the matrix gathers, of displays 51, 55, 56, and 57represents the ratio of selected measured potential and current valuessystematically generated by the array 21 at logging locations A . . . Din FIG. 4.

As can be seen in FIG. 5, indexing is as follows: each display 51, 55,56, and 57 comprises a 5×5 matrix of impedance values. Rows of thematrix are denoted by R1, R2 . . . R5, while columns are specified byC1, C2 . . . C5. Increasing order of the matrix entries is fromleft-to-right and top-to-bottom, respectively. That is to say, thecolumns C1, C2 . . . C5 increase from left-to-right as a function ofordered increased in depth of the current electrode and each of thecolumns C1, C2 . . . or C5 has a common current electrode numericalindentifer and a common depth identifier associated with the indexnumber of the common current electrode of each column. Additionally,each column is identified with a sequentially increasing internalindexing number of the potential measuring electrodes of the array.

Within each of the rows R1, R2 . . . R₅, it is seen that the entriesincreases in depth from left-to-right as viewed, viz., increasing fromthe smallest depth value for the entry at the left of the rows, asviewed, ending with the largest depth value at the right of the display.Note also that each row R1 . . . R5 has a common potential electodenumerical identifier associated within each such row but formrow-to-row, such identifier sequentially increases from top to bottom asviewed.

As a result of indexing the measured impedance values as set forth inthe displays 51, 55, 56, and 57, the interpreter can easily synthesizeoperations of different types of logging arrays so as to provided amultiplicity of different current penetration patterns and hence moreeasily determine the resistivity changes as a function of lateraldistance into the formation under survey.

It should be noted that while the zone of resolution of each display 51,55, 56, and 57 is fixed by the active length of the array (and thus inFIG. 4 is equal to 5 logging stations), individual impedance entriesmaking up each display can be the average of several values since thelatter are associated with common logging stations. For example, inviewing the logging positions A, B, C, and D in FIG. 4, it is evidentthat there are two (2) logging stations for which precursor measurementsare obtained each time collection occurs, viz., stations d₄ and d₅.Hence impedance entries associated with these stations can be an averageof the four obtained precursor measurement, if desired, and that averageentry may be used in the resulting displays.

It should be further noted that while each of the displays 51, 55, 56,and 57 only represent a 5×5 matrix, in actual operations there are manymore entries per matrix gather. For example, if the last numberedelectrode assembly of the logging array is designated by the number "M", then there are M×M entries per matrix gather. However, note that onerow, usually the R1 row, will be populated by absolute impedance valuesand the remaining rows, viz., R2, R3 . . . RM will be composed ofdifferential impedance values determined by dividing the voltagedifference between adjacent pairs of potential electrodes of the loggingarray, and the energizing current. Because of the existence andoperability of the principle of linear superposition, the resultingmatrix has the attribute of being able to systemically relate anyarbitrary set of synthetic emitting currents of the logging array tocorresponding differential potential distribution on any selected numberof potential electrodes of the array in accordance with

    ΔV[d.sub.k +Na]=ΔZ[d.sub.k +Na]* J[d.sub.k +Na]

where (-) indicates a matrix or vector quantity; depth d_(k) +Nacorresponds to that of the scan depth number associated with aparticular matrix; and ΔV and J are M×1 column vectors given by:##EQU4##

Furthermore, thus ΔZ[d_(k) +Na] is the M×M modified formation impedancematrix gather akin to that set forth in displays 51, 55, 56, or 57 ofFIG. 5 when referenced to scan depth d_(k) +Na, and has the followingform: ##EQU5##

UPHOLE AND DOWNHOLE HARDWARE

In order to provide accurate control to the logging array, the presentinvention of FIG. 1 contemplates using surface control circuitry tomonitor downhole operations, that is, to used a controller withinprocessor-controller circuit 17 of van 18 at the earth's surface toclock operations of a slave controller within midsection 30 within thelogging array 21 of FIG. 1.

FIGS. 6 and 7 illustrates such operations in detail wherein up-holecircuitry is set forth in FIG. 6 and downhole circuitry in FIG. 7.

As shown in FIG. 6, circuit 17 includes a master clock 60 forcontrolling received/transmit circuit 61, master input logic circuit 63,and I/O circuit of digital computer 64 through timing logic circuit 65.Logging data of a formate and character to be described hereinafter,passes upward from logging array 21 through cable 12, over pulley 13 ofderrick 14. Next, the data is transmitted through hoisting unit 16 viaconductor 19 to the receive/transmit circuit 61 and thence through themaster input logic circuit 63 to digital computer 64. At the computer64, the data is displayed in real time at display unit 66 (say, to checkfor phase distortion) and then after calculations have been completed,the final matrix is recored at recorder 67. Since the logging data isinitially in a format that is incompatible with computer operation itmust be first demultiplexed at maste input logic circuit 63 (i.e.,changed from serial to parallel format) and then indexed with depthmarkers from depth encoder 62 attached to hoisting unit 16. To provideproper word and block lengths to the data compatible with processingwithin computer 64, the master logic circuit 63 is carefully controlledusing the timing logic circuit 65 in conjunction with master clock 60.

As to depth encoder 62, note that in operations such depth encoderprovides the absolute depth a reference location of the array 21relative to the earth's surface 15 (preferably based on the depth of theshallowest electrode assembly of the logging array at each stationarylogging position). The depth associated with measurements originating atthe remaining electrode assemblies of the array 21 of FIG. 1 isdetermined from the known spacing "a" between adjacent electrodeassemblies E₁ . . . E_(M).

In operation, the master clock 60 produces a series of timing (clock)pulses which are applied to logic circuit 65 and thence to the remainingcircuits to dictate when such circuits are to perform. Each operationusually requires a certain number of clock pulses, and consequently, thetiming to complete one of the various operations is an exact multiple ofthe clock pulses. For example, the readout of master input logic circuit63 is achieved during a specific interval of time that is an exactmultiple of the clock pulses form master clock 60. As one subset ofcircuits is disabled, a new subset is enabled by the time pulses so asperform new operations. In this regard note that format control unit 68is capable of manual changes during data transfers to compute 64. Inthat way the format of the data can be varied to meet differentinterpretational requirements as occurring at the real time display unit66 and at the data recorder 67 in the manner previously discussed.

As previously mentioned, FIG. 7 illustrates downholes circuitry indetail.

As shoen, a clock 80, in conjunction with timing logic circuit 81, isused to control operations of electrode assemblies E₁, E₂ . . . E_(M) ofthe logging array generally indicated at 21 in the FIG., in conjunctionwith and in response to the uphole timing circuitry ofprocessor-controller circuit 17. In more detail, assume that clock 80 isfree running, and an initialization signal 79 from the uphole circuitry17 appears at timing and control logic 81 and initiates operation. Alsoassume that by means of the start signal 79, initialization with theremaining circuit elements of the downhole circuitry has occurred. Thatis, using an initialization signal from timing and control logic 81; thefollowing elements (formatter/transmit buffer 85; A/D convertor 86; gaincontrol logic 87; and multiplexers 84 and 89) are correctly initializedto begin operations.

Simultaneously, counter 90, associated with current generator 91, isinitialized and operations begin to allow current injection via currentswitch 95 to particular electrodes Ec₁, Ec₂ . . . Ec_(M) of theelectrode assemblies E₁, E₂ . . . E_(M) in sequence.

That is to say, after current generator 91 is activated to injectcurrent into the formation, current intensity, as well as particularabsolute and difference potentials at the potential electrode assembliesE₁ . . . E_(M) are measured and passed to the data acquisition circuits.More specifically, the absolute potentials are passed to comparator 83avia multiplexer 84, while at the same time the absolute potential ofelectrode assembly E₁ and all difference potentials are passed tomultiplexer 89, A/D converter 86 and formatter/transmit buffer 85 viasample/hold circuits 88a, 88b . . . 88m using particular binary gainamplifiers of a series of such ampliers generally indicated at 97 and98. Phase measurements at any of the electrode assemblies can beobtained by means of comparators 83a via the output of multiplexer 84.Note that comparators 83b and 83c are permanently connected to theoutputs of amplifiers 97a and 97m, respectively.

In this regard, it should be recalled that the gain associated withthese downhole measurements is used to increase accuracy, i.e.,correctly indicated the magnitude of the detected measurements beingamplified by particular amplifiers 97 and 98. But to avoid overloadingthe individual amplifiers 97 and 98 gain must be varied in accordancewith the signal to be amplified. The gain of each amplifier iscontrolled by the gain control logic 87 on the basis of the magnitude ofthe signal during the previous measurement cycle. The gain informationfrom each amplifier is passed to formatter/transmit buffer 85 togetherwith the output from the A/D converter 86 and becomes a part of thefinal data word.

Current Intensity

In FIG. 7, current intensity is seen to be measured via binary gainamplifier 100 whose gain is also controlled by gain control logic 87.For a useful current range of 500 microamperes to 10 amperes, theresistor 101 in the current path should not exceed 0.1 ohms, resultingin a voltage input to the amplifier 100 in a range from 50 microvolts to1 volt. Hence, its programmable gain is in binary steps ranging from 5to 100,000 and requires at least 15-bit gain code.

Phase Measurements

In oreder to obtain the phase of the potentials at the electrodesassemblies, the interval times between zero crossings of the signalversus the phase reference, i.e., the start of the current sine wave,are measured. The content of counter 90 serves as the phase referenceand is loaded into phase registers 92a, 92b, and 92c at the preciseinstant the comparators 83a, 83b, and 83c detect a zero crosssing of thecorresponding potential signals. The phase of the potential signal canbe determined at any of the electrode assemblies from the aboveinformation, however, in the present embodiment only the phase of thefar electrode assembly and the assembly at which the current is beingejected is desired. Any change in the counts indicated in a given phaseregister is directly proportional to phase distortion providing a directindication of reliability of the associated measurements.

Absolute and Differential Potential Measurements

In order to provide a current intensity J at electrode Ec₁ of theelectrode assembly E₁, the following must occur in sequence: first,counter 90 is reset via a reset signal from timing and control logic 81.Clock pulses at the input of counter 90 increment its content until acomplete current cycle is generated. The sine lookup table D/A converter102 then converts the content of counter 90 to produce a series ofdiscrete current values whose individual amplitudes vary sinusoidallywith time. After amplification via amplifier 103, the sinusoidallyvarying current is gated through current switch 95 to the currentelectrode Ec₁ of the electrode assembly E₁ and thence into the adjacentformation in the manner previously described. In this regard, it isassumed that the electrode assemblies that make up the generic loggingarray now being described have essentially infinite internal impedancesso that they do not draw appreciable current from the surrounding mediumand they are physically small ringlets as previously described so thattheir presence does not tend to alter significantly the potential fieldin the vicinity of the outer surface of the array. Additionally, thecurrent from the current electrode, of course, must return to close themeasuring circuit and this is done by means of remotely located returnelectrode 28 of FIG. 1. The return electrode 28, for all practicalpurposes, appears to be located at infinity.

The absolute voltage at common electrode assembly E₁ is measured withrespect to a normalizing reference potential such as provided byreference electrode 29 of FIG. 1, along with all differential potentialsat all adjacent pairs of potential electrodes Ep₁, Ep₂ . . . Ep_(M) ofthe electrode assemblies E₁, E₂ . . . E_(M) comprising the array 21.That is to say, the absolute potential at potential electrode Ep₁ ofassembly E₁ is indicated via amplifier 97a, while potential differencesof adjacent pairs of assemblies E₂, E₃ . . . E_(M) are measured by meansof differential binary gain amplifiers 98b, 98c . . . 98m.

Thereafter, the collection process is repeated using gating circuit 95to activate the next adjacent electrode assembly with current, viz., thecurrent electrode Ec₂ of assembly E₂. Absolute potential is measured atcommon electrode Ep₁ using amplifier 97a, while potential differences ofthe adjacent pairs of assemblies E₂, E₃ . . . E_(M) are measured bymeans of amplifiers 98b, 98c . . . 98m in similar fashion as discussedabove. After all electrode assemblies have been current activated asdiscussed above, the array is moved one logging station--either up ordown--, and the entire collection process is repeated.

It should be noted that as the collection cycle is repeated, at theearth's surface 15, the measurements are annotated and then processed toprovide impedance entries of a series of finite, overlapping matrixgathers associated with a series of depth intervals. That is to say,each gather of impedance entries is indexed to a selected mid-centraldepth interval dependent on the active length of the electrode array(between electrodes E₁ and E_(M)) that defines the shallow and deepdepth limits of each such gather.

It remains now to indicate briefly how the data may be measured and thentransmitted uphole based on data records divided into words and blockscompatible with the computer 64. A brief description of the collectionand transmission format is in order and is set forth below.

COLLECTION, TRANSMISSION AND TAPE FORMAT

A data record (also referred to previously as a matrix gather) consistsof depth information provided by the depth encoder 62 of FIG. 6 followedby data collected by the array 21 of FIG. 1 as outlined previously.

More specifically, with the tool stationed at a preselected depth, depthinformation is input to digital computer 64 via master input logic 63,and a start signal to the tool is transmitted via receive/transmitcircuit 61. As previously described, this start signal initiates thedata collection process whereby (i) absolute potential data, (ii)difference potential data, (ii) current intensity data, and (iv) phasedata are transmitted sequentially via formatter/transmit buffer 85 in apredetermined sequence to controller-processor circuit 17, as indicatedat FIG. 7.

Each data word consists of the 16-bit output from the A/D converter plusa maximum of 16 bits for the corresponding gain code. Measurements atthe electrodes are gain-indexed. Gains must be set in binary steps from10 to 3500 for absolute potential measurements (requiring a 9-bit gaincode); from 10 to 200,000 for potential difference measurements(requiring a 15- bit gain code); and from 5 to 100,000 for currentmeasurements (requiring a 15-bit gain code). The phase measurement doesnot require a gain code.

The amount of data contained in a data record as outlined above isdetermined by the number of active electrode assemblies on the tool. Forexample, assuming that 73 active electrode assemblies are utilized, thiswill result in 1 absolute potential, 72 difference potentials 1 currentintensity and 2 phase measurements, and correspondingly, 74 gain codes,giving a total of

    (76+74)*(73)*(16)=175,200 bits

of information per data record.

The present invention does of course not preclude the possibility ofcollecting the above-mentioned data record repeatedly with the toolremaining at the current depth station in order to allow the use ofsignal enhancement processing methods to reduce the effects ofenvironmental noise on the quality of the impedance elements.

Simultaneously with the real-time processing of the acquired data, it ispossible to transfer the information to a storage device such asmagnetic tape for later access in processing. The particular format thatone chooses for this purpose is somewhat arbitrary but should possiblybe dictated by convenience with respect to characteristics of therecording device and the nature of the subsequent processing.

For comprehensive identification purposes, data records are preceded byheader information which, in addition to the usual informationdescribing time, site, etc., should also contain data describing suchparameters as electrode spacing "a", mandrel diameter, number of activeelectrodes used in collecting the data, potential reference electrodelocation, etc.

Now after briefly describing the format of the collected and transmittedlogging data, the annotation aspects of the method of the invention havebeen simplified.

(1) Since the electrode assemblies E₁ . . . E_(M) are bifurcated formingfirst and second ringlets, they essentially occupy the same depthcoordinate. That is to say, the axial distance between the ringlets isso small compared to the distance between adjacent electrode assemblypairs that for purposes of discussion they occupy the same general depthlocation as previously defined.

(2) In order to manipulate the quantities as functions of severaldifferent variables in a rapid manner, the electrode assemblies E₁, E₂ .. . E_(M) are numbered using the indexing function previously defined,beginning at the top of the mandrel and ending at the deepest point ofthe mandrel in the borehole. The mid-central electrode assembly isnumbered N+1 and deepest positioned electrode assembly is numbered2N+1=M. Accordingly, once M is established, the numbering of theelectrode assemblies is straight forward. That is, assume that M=73;hence N+1 is 37 and the electrode assemblies above the mid-centralassembly would be E₁, E₂ . . . E₃₆ and those below would be E₃₈ . . .E₇₃.

(3) The measured potential quantities are also indexed, along the linespreviously described. That is to say, with the electrode current beingemitted from a particular assembly at depth d_(k) +(p-1)a where p is 1,2, 3 . . . M, the absolute and differential potentials of assemblies E₁. . . E_(M) are indexed based on the internal numbering order of theactive assemblies as previously described.

Note that there are M-1 potential difference measurements while there isonly one absolute voltage measurement each cycle.

From the above obtained current and potential values, their impedanceratios can be readily determined, and indexed as a function currentelectrode position and potential electrode position, as previouslydescribed.

(4) The next step is to repeat the above-described measurements.Indexing of such measurements is as previously stated.

By continuing to displace the array upward or downward along theborehole, one logging station at a time and with the array stationaryrepeating the collection process, it is seen that a series of impedancemeasurements can be developed indexed to current position, and potentialelectrode position. Next, these values can be reindexed to form a matrixgather of impedance entries associated with formation adjacent to thestationary array of FIG. 1 normalized to depth of the zone of traverseof the matrix, the scan station number as well as the depth of themid-central electrode assembly.

It should be noted that because of the principle of linear superpostionthe impedance matrix gathers provided by the method of the presentinvention have the attribute of being able to relate an arbitrary set ofcurrents issuing from the M electrodes to the corresponding absolutepotential and potential difference distribution appearing on the Melectrodes, with respect to the remote reference electrode as previouslymentioned.

As a final matter, note the impedance gathers of the present inventioncan systematically comprise an extremely large number of entries, viz.,if the last electrode is designated "M" then there are M×M entries.However only one row consists of impedance entries derived from absolutepotential measurements at the electrode assemblies of the array.Accordingly the greater number of entries consists of impedance valuescalculated from potential differences between adjacent pairs ofelectrode assemblies, viz., equal to M-1 rows of entries. These entriesdefine a modified formation impedance matrix indexed as previouslydescribed.

END USE OF THE DATA

In order to accurately determine resistivity of the formation as afunction of lateral distance from the borehole, assume that the improvedimpedance entries of each matrix gather have been determined inaccordance with the steps set forth above. Also assume that thereciprocal matrix comprising each gather has been generated. Then basedupon the superposition principle, any arbitrary set of voltagedifferences between adjacent electrode assemblies to a correspondingcurrent response can be related by such determined reciprocal impedancematrix. That is, for a series of drive voltage differences between aselected number of electrode assemblies simulating different focusedtool arrays, corresponding current responses can be expressed as asystem of linear equations wherein terms thereof include column vectorsΔV, J and the reciprocal ΔZ⁻¹ matrix, viz.:

    J=ΔZ.sup.-1 ΔV

where

    ΔV.sup.T =[V.sub.1 (V.sub.2 -V.sub.1)(V.sub.3 -V.sub.2) . . . (V.sub.M -V.sub.M-1)]

    J.sup.T =[J.sub.1 J.sub.2 . . . J.sub.M ]

second ΔV^(T) and J^(T) denote the transposes of vectors ΔV and J,respectively.

While the prior art has suggested (i) how to use the reciprocals ofrelationships such as shown above, viz., how to relate a series ofsynthetic voltages calculated by productizing a set of current vectorswith the resistance matrix of the formation under study (see Stelzer etal) and (ii) how to construct an equivalent network simulating an earthformation involving the reciprocal resistance matrix to aid incalibrating electric tools (See THE LOG ANALYST, May-June 1979, "ATheory of Equivalent Artificial Networks Simulating The SubsurfaceFormations, and Their Application In Electric Well Logging", ZoltenBarlai), none as far as we are aware have contemplated a method forsystematically relating a determinable formation characteristic otherthan synthetic voltage patterns, in general or in particular, for use indetermining a formation characteristic (or series of characteristics) asa function of mud filtration invasion and differing resistivitycontrasts in and around the borehole.

For example, in Stelzer, op. cit, the apparent resistivity of a selectedsynthetic array is determined by linear combination of the normalizedsynthetic voltage values divided by a quantity productized with a linearcombination of the reciprocals of the absolute effective lineardistances among the several electrodes. Such results occur only afterthe synthetic voltages have been calculated based on productizingdifferent sets of current patterns with a previously determinedresistance matrix of the formation under study. Moreover, assuming theeffective radii of the electrodes of the logging array are negligible,then resistivity can be said to be a direct function of the syntheticvoltage entries. But experience indicated that often other factors havea strong effect on array response especially where significant mudfiltration invasion and large differing resistivity contrasts in andaround the borehole are present.

Fig. 8 is an equivalent artificial terminal network that uniquelysimulates an earth formation over an equivalent length equal to Mlogging stations along a borehole penetrating the formation under studyand illustrates the unique correspondence of the admittances of thatnetwork and the entries of each reciprocal impedance matrix gather asprovided by the electrode array and uphole circuitry of FIG. 1, evenwhere mud filtrate invasion of the formation is significant. That is tosay, FIG. 8 shows that each reciprocal impedance matrix gatherrepresents an equivalent terminal network that accurately simulates theearth formation under study even under difficult logging conditions.

As shown, the network comprises an array of lumped admittances 140 and144 connecting terminal P₁, P₂ . . . P_(2N+1) with each other and withground terminal 145, respectively. The terminals of this networkcorrespond to the electrode assemblies of the logging array previouslydescribed, with P₁ corresponding to electrode assembly E₁ and P₂ N+1corresponding to electrode assembly E_(M). The total number of distinctadmittances in the network is (N+1)(2N+1). It follows that if there areM electrode stations over the zone of interest, then there are M(=2N+1)lumped admittances interconnecting the terminals to ground terminal 145.

If arbitrary current of intensities J₁, J₂ . . . J_(M) are applied atthe terminals P₁, P₂ . . . P_(2N+) 1 and corresponding node voltages aredefined as V₁, V₂ . . . V_(M), then application of Kirchoff's law ateach terminal provides the following set of equations: ##EQU6## wherethe definition Y_(i),j =Y_(j),i is implied.

This set of equations can be rearranged in an obvious manner to yieldthe following system: ##EQU7## where the diagonal terms of the matrixare obtained by summing all connecting admittances from the terminalcorresponding to the given row (or column), that is,

    Y.sub.p =Y.sub.p,1 +Y.sub.p,2 +. . . =Y.sub.p,M, p=1, 2 . . . M.

For simplicity and clarity of notation in what is to follow, let thecoefficients Y_(i) and Y_(i),j be further replaced by X's according tothe following scheme, i.e., let

    X.sub.i,j =-Y.sub.i,j, i not equal to j

    X.sub.i,i =Y.sub.i

so that the above system of equaitons can be rewritten conveniently asfollows: ##EQU8##

This set of equations relates the absolute voltages; however, it isdesired to put them in a form involving difference voltages betweenadjacent terminals. Toward this end, and following the notation usedearlier, it is possible to express these difference voltages ΔV_(j)according to the relations

    ΔV.sub.j =V.sub.j -V.sub.j-1, j=2, 3 . . . M

    V.sub.p =V.sub.1 +ΔV.sub.2 +ΔV.sub.3 +. . . +ΔV.sub.p, p=2, 3 . . . M.

This in turn allows the above system of equations relating the absolutevoltages and currents to be rewritten as follows: ##EQU9##

The preceding system of equations gives the desired result, i.e., inmatrix notation we have the following form:

    J=U ΔV

where

    V.sup.T =[V.sub.1 (V.sub.2 -V.sub.1)(V.sub.3 -V.sub.2) . . . (V.sub.M -V.sub.M-1)]

    J.sup.T =[J.sub.1 J.sub.2 . . . J.sub.M ]

and the elements of the matrix U are simple, unique linear combinationsof the lumped parameter circuit admittances Y_(i),j (through the X's).

Therefore, assuming the inverse of the modified impedance matrix ΔVprovided by the logging array of FIG. 1 exists, then from the relation

    U=ΔZ.sup.-1

the admittances Y_(i),j can be uniquely determined to bring theequivalent circuit into a one-to-one correspondence with the reciprocal(inverse) of the modified impedance matrix.

This means that discussion of the various properties of the loggingarray can be carried out in terms of the equivalent network. That is, inaccordance with the present invention the perfect and uniquecorrespondence of the admittance matrix of the network of FIG. 8 and thereciprocal impedance matrix provided by the apparatus of FIG. 1, can beused in the following manner to provide a systematic method ofdetermining a series of response characteristics whereby filtrateinvasion and differing resistivity contrasts in and around the boreholeare taken into account. That is to say, the present invention describesa method of synthesizing the true response characteristics of acombination of different hole-centered electric logging tools in avariety of difficult borehole conditions as provided by (1) determiningimpedance values of an earth formation penetrated by a borehole filledwith a drilling mud of resistivity (Rm), and (2) selectivelymanipulating the impedance values as impedance entries of a reciprocalmatrix so as to synthesize operations of different hole-centered toolsover an associated depth increment with suprising accuracy, wherein eachof said reciprocal impedance matrices is associated with a matrix gatherindexed to one of a series of finite, overlapping depth scan incrementsof the formation measured along the borehole. Each scan increment isdependent on the array length L of the electrode array to define shallowand deep depth markers as well as being centrally indexed to the depthin the borehole of a mid-central electrode assembly of the array at thetime of data collection.

In order to provide a true indication of the formation resistivity (Rt)even though the formation is interspaced from the borehole by an invadedzone of resistivity (Rxo) of unknown lateral extent due to drilling mudfiltrate invasion, the method of the present invention specifies thefollowing steps:

(i) first, an array of hole centered M electrode assemblies of equalincremental electrode spacing "a", is calibrated to obtain sets ofcalibration factors normalized to known current response and voltageinitiation patterns in a known resistivity zone of response, saidelectrode assemblies having a known internally ordered numbering index,and each comprising a current electrode and a potential electrode, saidsets of calibration factors each being addressable as a function ofborehole conditions including said difficult borehole conditions as wellas by synthetic computer focused array type;

(ii) next the array is positioned in the borehole, wherein the absolutedepth of at least one electrode assembly is continuously known withrespect to a predetermined depth datum level measured from the earth'ssurface;

(iii) while the array is stationary within the borehole, current, of aknown value, is injected from each of the assemblies through thedrilling mud of resistivity (Rm) and the invaded zone of resistivity Rxoand thence into the formation of resistivity Rt;

(iv) during current injection from each assembly, the absolute potentialat a common assembly as well as the potential differences betweenadjacent pairs of potential electrodes, are measured in a rapid mannerwherein the potential measurements can be indexed to common loggingstations of a series of equi-spaced logging stations located along theborehole having a spacing incremental distance of "a" where "a" is thedistance between electrode assemblies;

(v) then impedance values from the measured absolute and differencepotential values and their associated injection currents, arecalculated, each value being indexed to said known internal indexingnumbers of active current and potential electrodes used in themeasurements;

(vi) next the impedance values are reindexed into impedance entries of aseries of overlapping modified matrix gathers ΔZ, each gather ΔZ beingassociated with a predetermined segment of said formation equal invertical extent to M logging stations, and comprising M×M impedanceentries where M is the largest number of the numbering index of theelectrode assemblies comprising said array and in which the ratio of thenumber of difference impedance entries to absolute entries is aboutM-1:1,

(vii) thereafter, each modified matrix ΔX is inverted to form a modifiedreciprocal matrix thereof ΔZ⁻¹ in accordance with conventional matrixinversion techniques;

(viii) then computer focused response parameters are generated using thereciprocal matrix ΔZ⁻¹ of step (vii) in conjunction with the samecurrent response and voltage initiation patterns of step (i);

(ix) finally, sets of calibration factors of step (i) are searched untilthe products of a praticular set of calibration factors and the responseparameters of step (viii) for all synthetic tool arrays are essentiallya constant whereby the difficult borehole condition is deduced even inthe presence of high true resistivity to mud resistivity contrasts andirrespective of the fact that synthetic sets of potential patterns havebeen used as initiators of the subsequently generated computer focusedresponse parameters.

Having briefly described the invention in the manner set forth above,the steps requiring additional comment will now be discussed in moredetail below.

STEP (i)

In this step, note that in order to calibrate an array of hole centeredM electrode assemblies to a series of calibration factors indexed todifferent borehole conditions and array response, the number ofelectrode assemblies of the array must be matched to that used incollecting the logging data in the field. In this instance, M is assumedto be equal to 73, spacing factor "a" is 5 inches, mandrel diameter is3.75 inches, and borehole diameter is 8 inches. Hence total active arraylength L is 30 feet. Additionally, different resistivity contrasts andvarious different filtrate invasion distances must be assumed and setsof calibration factors must be genetated with such factors being indexedto different borehole conditions, as well as to normalized currentresponse and voltage initiation patterns.

Specifically, a series of formation to mud resistivity contrasts isfirst assumed in an eight inch borehole along with known invasionconditions and characteristic responses of synthetic tool arrayscomputed by applying appropriate solution tecnhiques to the well-knownboundary value problem describing current and voltage patterns inconducting media. In this regard, see V. N. Dakhnov, (1962) "GeophysicalWell Logging," translated by G. V. Keller, Quarterly of the ColoradoSchool of Mines, Vol. 57, No. 2, Chapters 3 and 4.

Next, sets of calibration factors are determined so the product of thecalculated responses and the set of calibration factors is equal to thedesired characteristic resistivity which may generally be taken as Rt.

If it is assumed in this regard that the calibration factors aredesingated by the symbol k_(q), then the results set forth above aredefined by a series of relations of the form

    Ra.sub.q =k.sub.q R.sub.q, q=1, 2 . . . N+1

where Ra_(q) now represents the apparent resistivity response of theassumed formation.

The calibration factors can then be determined by setting

    Ra.sub.q =Rt, q=1, 2 . . . N+1

for example, to give the desired resistivity response for the particularformation.

In accordance with the present invention, the desired synthetic toolarrays can be described by assuming that the network of FIG. 8 simulatesthe formation under study, and that any given tool array to besynthesized is positioned and operated so that initially the mid-centralelectrode assembly is chosen as the current measuring electrode, i.e.,at mid-central assembly (N+1). The intensity of the current within theadjacent formation of interest is determined when selected sets ofpotentials exist at the terminals P₁, P₂ . . . P_(M) of the circuit ofFIG. 8.

The depth dependent responses thus initially will be characterized bythe ratio V_(N+1) /J_(N+1), and the current J_(N+1) will be determinedfrom the following relationship:

    J.sub.N+1 =Y.sub.N+1,1 (V.sub.N+1 -V.sub.1)+. . . +Y.sub.N+1,.sub.N+1 V.sub.N+1 +. . . +Y.sub.N+1,M (V.sub.N+1 -V.sub.M).

Note, thus, that the above determination of the current J_(N+1) involvesonly the admittance connecting the (N+1)th terminal to the groundterminal 145 of FIG. 8, and the admittances between (N+1)th terminal andthe remaining terminals. Hence various admittances or combinationsthereof can be isolated by imposing appropriate voltage distributions onthe electrode assemblies of the array.

For example, assume that a series of linearly independent potentialvectors S₁, S₂ . . . S_(N), S_(N+1), are imposed on the electrodeassemblies of the logging array having the distribution depicted inTable III.

                  TABLE III                                                       ______________________________________                                         ##STR3##                                                                              ##STR4##                                                                                ##STR5##                                                                                ##STR6##                                                                              ##STR7##                                 ______________________________________                                         LEGEND:                                                                        - S.sub.1 contains 1 unit potential value                                     - S.sub.2 contains 3 unit potential values                                   .                                                                             .                                                                             .                                                                               -S.sub.N contains 2N - 1 unit potential values                               - S.sub.N+1 contains 2N + 1 unit potential values.                      

As shown, each vector is symmetrical about assembly N+1 and hasindividual unit amplitudes of either ZERO or ONE units(s). The number ofONE'S in any M×1 column of any vector S_(p) is in accordance with

    2p-1, where p=1, 2 . . . N+1.

In other words, whereas vector S₁ has only a single potential ofamplitude ONE (at assembly N+1) for a boxcar distribution of 1×1 aboutthe assembly N+1, the vector S₂ has a distribution length of threedefining a boxcar distribution of 3×1 about the same assembly, whilevector S₃ has a boxcar distribution of 5×1 at the same place.

On the other hand, for vectors S_(N), and S_(N+1) where N=36 if M is 73,the boxcar distributions are 71×1 and 73×1, respectively, as indicatedin Table III.

As each of the characteristic potential vectors of Table III is applied,one at a time, starting with the S_(N+1) vector, corresponding currentvalues can be sequentially determined for the normalized synthesizedelectrode array. For purposes of annotation in accordance with thepresent invention, the resulting current component is called the measurecurrent for the correponding potential vector of the designateddistribution, viz., called J(S_(q)) where "q" indicates the potentialvector from which the current is determined. Hence to calculate themeasure current J_(N+1) starting with the S_(N+1) vector, i.e., foridentical ONE voltages on all electrode assemblies, the (N+1)th row ofthe productized admittance matrix and the S_(N+1) potential vectoryields,

    J(S.sub.N+1)=Y.sub.N+1,N+1

where J(S_(N+1)) denotes that the measure current is in response topotential vector S_(N+1). The quantity Y_(N+1),N+1 can be referred to asthe self-admittance of the (N+1)th electrode assembly in the particularformation of interest.

Similarly, for the potential distribution described by S_(N), the(N+1)th row of the productized admittance matrix and the vector S_(N)yields

    J(S.sub.N)=Y.sub.N+1,1 +Y.sub.N+1,N+1 =Y.sub.N+1,M.

And still further, for vector S₋₁ the (N+1)th row of the productizedadmittance matrix and the S_(N-1) potential vector equals

    J(S.sub.N-1)=Y.sub.N+1,1 +Y.sub.N+1,2 +Y.sub.N+1, N+1 +Y.sub.N+1,M-1 +Y.sub.N+1,M.

The pattern is easily discerned: that is, each measure current is simplythe sum of the self-admittance of electrode N+1 plus all admittancesconnecting it to other electrodes having zero potential (based on ZERO'sin the distribution order of the potential vector). It easily followsthat the general expression for the measure current is given by##EQU10##

Since the distribution of the characteristic potential vector is suchthat all nonzero components have a ONE value, the reciprocal of themeasure currents J(S_(q)) can be equated to resistance of the earthformation for which synthesization is occurring, viz., so that

    R.sub.q =1J(S.sub.q)

and

    R.sub.1 <R.sub.2 <R.sub.3 . . . R.sub.N-1 <R.sub.N <R.sub.N+1.

The resistance inequalities are based on the fact that with increasingindex q of the resistance R_(q), which represents the response of theqth synthesized tool array, fewer current paths are involved along withfewer short length paths so that deeper lateral response results. Thatis to say, the increasing orders of R_(q) represent successive,increasing orders of deeper response within the formation being modeled.Furthermore, for a series of R_(q) 's, sets of response parameters canbe generated for which sets of calibration factors k_(q) can in turn, becalculated. Result: the interpreter is provided with a systematic methodby which formation resistivity and depth of invasion of drilling mud inand around the borehole based on such characteristics, can be easilydetermined.

FIG. 9 illustrates graphically the relationship between increasingindexing numbers of synthesized tool array response and depth ofresponse from within the surrounding formation.

As shown, pseudo-geometic factor G_(q) for the various synthetic arraysis plotted as a function of diameter of invasion (Di) of a mud filtrateaccording to the voltage initiation patterns set forth in Table III. Asimple step-profile separates the invaded zone (Rxo) from the uninvadedzone (Rt), as is customary in calculations of this type.

In this regard, pseudo-geometical factor G_(q) is defined in the usualmanner by the relation

    G.sub.q =(Ra.sub.q -Rt)/(Rxo-Rt)

where the terms Rt and Rxo are as previously defined. The term Ra_(q)denotes the apparent resistivity of the synthesized arrays aftercalibration factors have been chosen. In this example, for purposes ofconstructing the curve of FIG. 9, constant multipliers k_(q) wheredetermined such that the apparent resistivity of each array could bemade equal to Rt in the situation for which Rt/Rm ratio is 100/1, withan 8-inch borehole and no invasion.

The curves of FIG. 9 clearly show the tendency of the computer-generatedresponse to be depth selective, with the more slowly increasing curves150a, 150b . . . 150d characterizing the deeper responses, and thecurves 150p, 150q . . . 150z characterizing the shallow responses. Notethat the curves 150a and 150s approximate the responses of the deepLaterolog (LLd) and the shallow Laterolog (LLs), respectively, logscovered by marks owned by, as well as being the product of focused toolsserviced by Schumlumber Inc., Houston, Tex., presently in common use inthe electric logging art. Such curves 150a and 150s, together with theadditional curves of FIG. 9 demonstrate the superiority in coverage andresolution of the method of the present invention.

FIGS. 10-28 are plots of calibration factors versus a series ofdifferent resistivity contrasts illustrating how sets of such factorsare unique to a selected borehole condition.

In this regard, the curves of FIG. 10 cover a very simple case whichserves to illustrate the concept. In the FIG. are depicted a series ofcurves 170 of logarithmic calibration factor (K_(q)) versus thelogarithm of the ratio Rt/Rm. The model formation on which thecalculations are based consists simply of an 8-inch borehole in anotherwise homogeneous medium. That is, there is no invasion.Furthermore, the array length is 30 feet and the electrode assemblylength is M, where M=73.

In more detail, curves 170 indicate that there exist sets of calibrationfactors, such as sets 170a, 170b, 170c . . . that can be uniquelyassociated with formation parameters of interest. For this very simpleillustrative case, of course there is only one parameter of interest,namely Rt, which can be uniquely determined if the mud resistively Rm isknown. Note also that the right-hand ordinate is labeled with integers 1through 37 representing the shallowest reading and deepest reading toolsof the previously described arrays that have been syntheticallysimulated using the patterns of response and initiation set forth inTable III. The calibration factors, K_(q), have been calculated based onthe following assumption: If the raw tool responses, i.e., the R_(q) 'sare given in a known resistivity contrast then their product with k_(q)'s (which correspond to the known resistivity contrast) will provideapparent resistivities Ra_(q) which area exactly equal to thecorresponding Rt in an 8-inch borehole. Clearly then, if raw data areavailable for an unknown condition, that is if R_(q) 's are availablefrom an earth formation which is thought to be uninvaded, for thissituation one need only search the curves of FIG. 10 to determine thatset of calibration factors which causes the resulting apparentresisitivities to all be equal to some constant value, thereby deducingthe borehole condition, i.e., Rt.

In this regard, it is seen that beyond contrasts of 1/1 (log contrastentry equal to 0.0), the responses of the deeper reading computerfocused tools say for tool indices of 10-37, have calibration factorsthat are not strongly dependent on contrast. On the other hand, theresponses of the shallow reading computer focused tools indicated thepresence of calibratin factors which are strongly dependent on contrast.

FIGS. 11-28 indicated further the interpretational aspects of thepresent invention, say under borehole condition involving increasingdegrees of mud filtrate invasion.

If it is desired to account for simple invasion in the formation modelon which the calibration factors are to be based, then the number ofparameters is increased, i.e., the calibration factors now will dependon not only Rt, but also on Rxo, Di, and Rm, for fixed boreholediameter. For such a situation the curves of FIGS. 11 through 28 arerepresentative, being plots of calibration factor (logarithmic) versusthe logarithm of the ratio Rt/Rxo for various diameters of invasion(step-profile). Note the ratio of Rxo to Rm is fixed at 10/1. Suchcurves then represent a data base which can be used in the mannerindicated above to deduce the invasion characteristics of the formation.That is, the borehole condition is deduced by searching the calibrationfactors to determine that set which, when multiplied by thecorresponding raw responses, gives apparent resisitivities which are allequal to the same constant value, the calibration factors having beendetermined in such a way as to yield the value of Rt in such acircumstance.

Somewhat more generally then, the method assumes the generation of adata base in which sets of calibration factors associated with differentborehole conditions (akin to sets 170a, 170b . . . of FIG. 10, or moregenerally sets of the type contained in FIGS. 11 through 28) have beencalculated for a particular configured logging array by which impedancematrix entries are to be obtained in the field. In this regard, the73-electrode array has previously been indicated as being the normaltool configuration for field operations. Assume also that such sets ofcalibration factors can be stored in a series of factor tables indexedby tool indices 1-37 associated with unit boxcar patterns of 1×1, 3×1,5×1, 7×1 . . . 71×1 and 73×1, respectively, within the digital computerof the controller-processor 17 of FIG. 1. Such tables are indexed suchthat entries in the tables are annotated by tool configuration andcurrent response and voltage initiation pattern, i.e., wherein toolconfiguration has been fixed in accordance with distribution of thepotential vectors about the array, in the manner previously discussed.To repeat, such table entries, viz., the calibration factors, must takeinto account a variety of different borehole and formation conditions,say under varying invasion conditions as well as under non-invadedconditions previously mentioned.

STEPS (ii)-(vi)

After the series of calibration factor tables has been generated whereinthe sets of calibration factors are addressed and annotated as set forthabove in accordance with the method of the present invention, thelogging array of the present invention is located in the borehole 8 ofFIG. 1. Next, logging of the formation in the manner previouslydescribed occurs resulting in a series of modified matrix gathers ΔZ'sbeing generated, each being associated with a predetermined segment ofthe earth formation under study. As previously indicated each matrixgather comprises M×M entries where M is the largest numbered electrodeassembly of the array. In the example of a 73-electrode array, M isequal to 73.

STEPS (vii)-(ix)

After the modified matrix gathers have been determined, each is invertedto form a modified reciprocal matrix gather ΔZ⁻¹ using conventionalmatrix inversion techniques. For example, experience has indicated thatstandard, well-known equation solving techniques such as GaussianElimination are sufficient to satisfactorily solve the problem ofinversion. In this regard, see "Linear Algebra And Its Applications",Gilbert Strang, Academic Press, 1976, for example.

Next, a series of computer generated responses is created using theobtained modified reciprocal matrices ΔZ⁻¹, normalized to particularcurrent response and voltage patterns similar to those used ingenerating the factor tables of step (i) as previously described. Thatis to say, using data provided of the field array, a series of R_(q)response values is generated, such response values being associated witha formation segment equal to M logging stations along the borehole whenlogging of the formation occurred. Finally, the obtained R_(q) valuesare compared with factor tables containing the sets of calibrationfactors. Searching is terminated (indicating the formation parameters ofinterest) when certain best selection critera are met. For example, fora given set of R_(q) values, the series of tables containing thecalibration factors are sequentially searched until the productizationof a predetermined set of calibration factors within a particular factortable provides the desired result, i.e., wherein the products of theR_(q) values, and the selected set of calibration factors k_(q) areconstant and equal for all 37 computor focused tool responses. If theoutput of the search results is in a plot format, with the Ra_(q) valuesto be plotted as the ordinate and tool index number 1-37 as theabscissa, then a best fit of data in accordance with the method of thepresent invention, occurs when a given set of calibration factors,k_(q), multiplied against the R_(q) values (=Ra_(q)) approaches ahorizontal straight line. But it should be emphasized that the tables ofcalibration factors must be augmented by the addition of other formationand/or borehole conditions. In this regard, additional tables can beannotated with other variables as set forth in FIGS. 11-28.

FIG. 11, for example, illustrates that a factor table can be easilyconstructed for an earth formation having thick beds with a mudresistivity of 0.1 ohm-meter (Rm), and an invaded zone resistivity of 1ohm-meter (Rxo). The calibration factors are plotted as the ordinate andresistivity contrast of the formation and invaded zone is plotted as theabscissa for an invasion diameter of 8 inches.

FIGS. 12-28 illustrate how even more factor tables for the same boreholebut different formation conditions than that of FIG. 11 can beconstructed wherein variations in diameter of invasion are taken intoconsideration, i.e., where filtrate invasion (D_(i)) is progressivelyincreased, viz., for 10 inches in FIG. 12 through 25 inches in FIG. 16and finally to 120 inches in FIG. 28. Note that for moderate to deepinvasion, the plots indicate a strong dependency of the depictedcalibration factors on the magnitude of resistivity contrast of theformation and the invaded zone, viz., Rt/Rxo. Also as the lateraldistance increases in the FIGS., the responses of the deeper readingcomputer focused tools play a progressively increasing role andsignificantly add to the number of calibration factors available in theproductization step and hence significantly increases the reliability ofthe final results. A key to any analysis: the fact that the particularsets of calibration factors for different tool arrays, in combination,can be uniquely tied to formation parameters of interest and thus can beused to provide a surprisingly accurate indication of such parameters.

It is acknowledged that calibration factors are old in the electriclogging art. For example, for the conventional focused tools thatprovide response curves 150a and 150s of FIG. 9, where calibrationfactors are chosen so that their responses are approximately equal tothe true formation resistivity (Rt) in uninvaded formations havingformation/mud resistivity contrasts in the range of 10/1 to 100/1normalized to an 8-inch borehole, it should be noted that such tools arehard-wired. Hence, once their calibration factors are determined, theyremain fixed unless a design modification necessitates change, as wherea change, for example, occurs in the spacing or size of the electrodesof such tools. But since operating characteristics in accordance withpresent invention are not design dependent but are readily calculatedusing the steps of the method of the present invention as set forthabove, the latter can be more easily associated with different boreholeconditions and hence offer a key to determining formation parameters ofinterest.

Although preferred embodiments of the invention have been described indetail, it should be understood that the invention is not limitedthereto as many variations will be readily apparent to those skilled inthe art and thus the invention is to be given the broadest possibleinterpretation within the terms of the following claims appended hereto.

What is claimed is:
 1. Method of synthesizing the true responsecharacteristics of a combination of different hole-centered electriclogging tools in a variety of difficult borehole conditions as providedby (1) determining impedance values of an earth formation penetrated bya borehole filled with a drilling mud of resistivity (Rm), and formingimpedance matrices with said impedance values, and (2) selectivelymanipulating the impedance values so as to synthesize operations ofdifferent hole-centered tools over an associated depth increment withsurprising accuracy, wherein each of said impedance matrices isassociated with a matrix gather indexed to one of a series of finite,overlapping depth scan increments of the formation measured along theborehole, each scan increment being dependent on the array length L ofan electrode array to define shallow and deep depth markers as well asbeing centrally indexed to the depth in the borehole of a mid-centralelectrode assembly of the array at the time of data collection, andthereby provide a true indication of the formation resistivity (Rt) eventhough the formation is interspaced from the borehole by an invaded zoneof resistivity (Rxo) of unknown lateral extent due to drilling mudfiltrate invasion, comprising the steps of:(i) calibrating an array ofhole centered M electrode assemblies of equal incremental electrodespacing "a" to obtain sets of calibration factors normalized to knowncurrent and voltage patterns in a known resistivity zone of response,said electrode assemblies having a known internally ordered numberingindex, and each comprising a current electrode and a potentialelectrode, said sets of calibration factors each being addressable as afunction of borehole conditions including said difficult boreholeconditions as well as by particular synthetic computer focused arraytype; (ii) positioning the array in the borehole, wherein the absolutedepth of at least one electrode assembly is known with respect to apredetermined depth datum level measured from the earth's surface; (iii)while the array is substantially stationary within the borehole,injecting current from each of said electrode assemblies--one at atime--into the adjacent formation; (iv) at each occurrence of currentinjection, measuring the absolute potential at a common electrodeassembly and the potential differences between adjacent pairs ofassemblies as a function of internal indexing numbers of active currentand potential sensing electrodes of the assemblies used in themeasurements; (v) advancing the array along the borehole an incrementaldistance equal to the electrode assembly spacing "a" between adjacentassemblies and repeating steps (iii) and (iv), again with the arraybeing substantially stationary within the borehole: (vi) re-repeatingsteps (iii)-(v) until the collection of all current and potential datahas been completed; (vii) calculating impedance values from the measuredabsolute and difference potential values and their associated injectedcurrents, each value being indexed to said known internal indexingnumbers of active current and potential electrodes used in themeasurements; (viii) reindexing the impedance values into impedanceentries of a series of overlapping modified matrix gathers, each gatherbeing associated with a predetermined zone of said formation coincidentwith depth of the mid-central assembly as the steps (iii) and (iv)occur, and comprising M×M impedance entries where M is the largestinternal indexing number of the array and in which the ratio of thenumber of difference impedance entries to absolute entries is aboutM-1:1 whereby each M×M matrix gather is surprisingly useful in deducingtrue resistivity of the formation even in the presence of high trueresisitivity to mud resistivity contrasts and irrespective of whether ornot synthetic sets of potential or sets of current values are later usedas initiators of subsequently generated responses of computer focusedsynthetic tool arrays; (ix) inverting each modified matrix gather toform inverted modified matrix gathers; (x) generating computer focusedresponse parameters using the same current and voltage patterns of step(i) and based on said inverted modified impedance matrix gathers; and(xi) searching sets of calibration factors of step (i) until the productof a particular set and the response parameters of step (x) for allsynthetic tool array types is essentially a constant whereby thedifficult borehole condition is deduced even in the presence of hightrue resistivity to mud resistivity contrasts and irrespective of thefact that synthetic sets of potential patterns have benn used asinitiators of the subsequently generated computer focused responseparameters.
 2. Method of claim 1 in which the array of M electrodeassemblies comprises assemblies E₁, E₂ . . . E_(M) and the internalindexing order thereof begins at assembly E₁ and ends at assembly E_(M)where E₁ is the shallowest assembly of the array and E_(M) is thedeepest assembly relative to the earth's surface.
 3. Method of claim 2in which the length L of the array comprising electrode assemblies E₁,E₂ . . . E_(M), is the vertical distance separating the shallowestelectrode assembly E₁ and the deepest assembly E_(M) along the borehole.4. Method of claim 3 in which the length L of the array also defines azone of formation that is related--by depth--to the calculated impedanceentries of each M×M matrix of step (viii).
 5. Method of claim 1 in whichthe entries of each M×M matrix comprise M columns and M rows ofimpedance entries wherein the M columns are each associated with acommon depth within the formation associated with a known currentelectrode.
 6. Method of claim 5 in which the known current electrode ofeach column of said M×M matrix is an internally numbered currentelectrode that signifies the common current electrode from whence thedrive current by which the impedance entry was calculated, originated.7. Method of claim 5 in which the M rows of each M×M matrix are eachassociated with a common internally numbered potential electrodesignifying the common potential electrode with which absolute anddifference potential values for calculating the impedance entries areassociated.
 8. Method of claim 6 in which each of the M columns of eachM×M matrix is associated with a potential electrode whose internalnumbering index sequentially changes in a known order from row to rowalong each column.
 9. Method of claim 7 in which each of the M rows ofeach M×M matrix is asssociated with a potential electrode whose internalnumbering index sequentially changes in a known order from column tocolumn along each row.
 10. Method of claim 5 in which M is
 5. 11. Methodof claim 10 in which the columns of the 5×5 matrix are made up ofabsolute impedance entries Z's and modified difference impedance entriesΔZ's and wherein the columns are provided with a sequentially increasingnumbering order that begins with 1 and ends with 5 and wherein each ofthe columns of the matrix has a common internally ordered currentelectrode from whence originated the drive current by which eachimpedance entry was calculated, and the rows of the 5×5 matrix areprovided with a sequentially increasing numbering order that begins with1 ends with 5 wherein each of the 5 rows has a common internally orderedpotential electrode from whence an absolute or difference potentialvalue associated with each impedance entry was determined.
 12. Method ofclaim 11 in which the impedance entries of the 5×5 matrix are eachidentified by a first subscript associated with the internally numberedpotential electrode from whence the absolute or difference potentialvalue by which the impedance entry was calculated, was originallyassociated, followed by a comma and then a second subscript associatedwith the internally numbered current electrode from whence originatedthe drive current associated with the impedance calculation.
 13. Methodof claim 12 in which the subscripts of the absolute impedance entries,Z's, and the difference impedance entries ΔZ's, of the 5×5 matrix are ofthe following order: ##EQU11## where: d₁ is the depth of the shallowestelectrode assembly of the array,d₂ is the depth of the next shallowestelectrode assembly E₂, d₃ is the depth of the mid-central electrodeassembly E₃, d₄ is the depth of the next to deepest electrode assemblyE₄, and d₅ is the depth of the deepest electrode assembly E₅.
 14. Methodof claim 13 in which a depth address of the internally numbered currentelectrode from whence originated the drive current by which eachimpedance entry was calculated is also provided each impedance entry.15. Method of claim 5 in which the columns of the M×M matrix are made upof absolute impedance entries Z's and modified difference impedanceentries ΔZ's and wherein the columns are provided with a sequentiallyincreasing numbering order that begins with 1 and ends with M where M isa whole number equal to the total number of electrode assembliescomprising the array, and wherein each of the M columns of the matrixhas a common internally ordered current electrode from whence originatedthe drive current by which each impedance entry was calculated, and therows of M×M matrix are provided with a sequentially increasing numberingorder that begins with 1 and ends with M wherein each of the M rows hasa common internally ordered potential electrode from whence an absoluteor difference potential value associated with each impedance entry, wasdetermined.
 16. Method of claim 15 in which the impedance entries of theM×M matrix are each identified by a first subscript associated with theinternally numbered potential electrode from whence the absolute ordifference potential value by which the impedance entry was calculated,was originally associated, followed by a comma and then a secondsubscript associated with the internally numbered current electrode fromwhence originated the drive current for the impedance calculation. 17.Method of claim 16 in which the subscripts of the absolute impedanceentries Z's, and the difference impedance entries ΔZ's, of the M×Mmatrix are of the following order: ##EQU12## where: d_(k) is thevertical distance from the earth's surface to the shallowest electrodeassembly E₁,"a" is the distance between electrode assemblies E₁, E₂ . .. E_(M), d_(k) +a is the depth of the next shallowest electrodeassembly, (d_(k) +[M-1]a) is the depth of the deepest electrode assemblyE_(M) of the array.
 18. Method of claim 1 in which step (i) is furthercharacterized by generating sets of calibration factors for differingarray combinations assuming an array configuration comprising M equallyspaced electrode assemblies driven by a series of different voltagepatterns, as a function of differing mud-to-formation resistivitycontrasts.
 19. Method of claim 18 with the additional step of storingthe generating calibration factors as a function of mud-to-formationresistivity contrasts and said differing array combinations.
 20. Methodof claim 19 in which the maximum number of stored sets of calibrationfactors are related to number of differing array combinations inaccordance with:

    k=(M+1)/2

where K is the maximum number of stored sets of calibration factors perarray combination.
 21. Method of claim 18 in which step (xi) includesmultiplying each set of calibration factors of step (i) on a one-by-onebasis, by individual response parameters of step (x) normalized tosimilar voltage initiation patterns of step (i) to form a series ofnormalizing values and selecting as a best fit a particular series thatcontain values of essentially constant magnitude.